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				<span dir="auto">Control Systems/Transfer Functions</span>
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				<div id="siteSub">From Wikibooks, open books for an open world</div>
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<p><font color="white"><small>The Wikibook of:</small></font><br></p>
<div style="font-size: 200%;"><font color="white">Control Systems</font></div>
<p><font color="white"><small>and Control Engineering</small></font></p>
</center>
</td>
<td><a href="http://en.wikibooks.org/wiki/Control_Systems" title="Control Systems"><font color="white">Table of Contents</font></a></td>
<td><a href="http://en.wikibooks.org/wiki/Control_Systems/All_Versions" title="Control Systems/All Versions"><font color="white">All Versions</font></a></td>
<td><a href="http://en.wikibooks.org/wiki/File:Control_Systems.pdf" title="File:Control Systems.pdf"><font color="white">PDF Version</font></a></td>
</tr>
<tr>
<td><a href="http://en.wikibooks.org/wiki/Control_Systems/Transforms" title="Control Systems/Transforms"><font color="white">← Transforms</font></a></td>
<td><a href="http://en.wikibooks.org/wiki/Control_Systems/Sampled_Data_Systems" title="Control Systems/Sampled Data Systems"><font color="white">Sampled Data Systems →</font></a></td>
<td><a href="http://en.wikibooks.org/wiki/Control_Systems/Glossary" title="Control Systems/Glossary"><font color="white">Glossary</font></a></td>
</tr>
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<table id="toc" class="toc">
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<td>
<div id="toctitle">
<h2>Contents</h2>
<span class="toctoggle">&nbsp;[<a href="#" class="internal" id="togglelink">hide</a>]&nbsp;</span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Transfer_Functions"><span class="tocnumber">1</span> <span class="toctext">Transfer Functions</span></a></li>
<li class="toclevel-1 tocsection-2"><a href="#Impulse_Response"><span class="tocnumber">2</span> <span class="toctext">Impulse Response</span></a>
<ul>
<li class="toclevel-2 tocsection-3"><a href="#Impulse_Function"><span class="tocnumber">2.1</span> <span class="toctext">Impulse Function</span></a></li>
<li class="toclevel-2 tocsection-4"><a href="#Step_Response"><span class="tocnumber">2.2</span> <span class="toctext">Step Response</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-5"><a href="#Convolution"><span class="tocnumber">3</span> <span class="toctext">Convolution</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Time-Invariant_System_Response"><span class="tocnumber">3.1</span> <span class="toctext">Time-Invariant System Response</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-7"><a href="#Convolution_Theorem"><span class="tocnumber">4</span> <span class="toctext">Convolution Theorem</span></a></li>
<li class="toclevel-1 tocsection-8"><a href="#Using_the_Transfer_Function"><span class="tocnumber">5</span> <span class="toctext">Using the Transfer Function</span></a>
<ul>
<li class="toclevel-2 tocsection-9"><a href="#Example:_Impulse_Response"><span class="tocnumber">5.1</span> <span class="toctext">Example: Impulse Response</span></a></li>
<li class="toclevel-2 tocsection-10"><a href="#Example:_Step_Response"><span class="tocnumber">5.2</span> <span class="toctext">Example: Step Response</span></a></li>
<li class="toclevel-2 tocsection-11"><a href="#Example:_MATLAB_Step_Response"><span class="tocnumber">5.3</span> <span class="toctext">Example: MATLAB Step Response</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-12"><a href="#Frequency_Response"><span class="tocnumber">6</span> <span class="toctext">Frequency Response</span></a></li>
</ul>
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<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=1" title="Edit section: Transfer Functions">edit</a>]</span> <span class="mw-headline" id="Transfer_Functions">Transfer Functions</span></h2>
<div class="PrettyTextBox" style="clear: right; margin:0 0 1em 1em; font-size:90%; width: 250px; padding:4px; spacing:0px;text-align:left;float:right;">This operation can be performed using this <a href="http://en.wikibooks.org/wiki/MATLAB_Programming" title="MATLAB Programming">MATLAB</a> command:<br>
<center><b><a href="http://en.wikibooks.org/wiki/Control_Systems/MATLAB" title="Control Systems/MATLAB">tf</a></b></center>
</div>
<p>A <b>Transfer Function</b> is the ratio of the output of a system to 
the input of a system, in the Laplace domain considering its initial 
conditions and equilibrium point to be zero. If we have an input 
function of <i>X(s)</i>, and an output function <i>Y(s)</i>, we define the transfer function <i>H(s)</i> to be:</p>
<p><br style="clear: all;">
<span id="Transfer_Function"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Transfer Function]</p>
</div>
<dl>
<dd><img class="tex" alt="H(s) = {Y(s) \over X(s)}" src="Transfer_Functions_pliki/392675f7d4a39f6ac2f9267d0ef08ab0.png"></dd>
</dl>
<p>Readers who have read the <a href="http://en.wikibooks.org/wiki/Circuit_Theory" title="Circuit Theory">Circuit Theory</a> book will recognize the transfer function as being the Laplace transform of a circuit's impulse response.</p>
<div class="center">
<div class="floatnone"><a href="http://en.wikibooks.org/wiki/File:Laplace_Block.svg" class="image"><img alt="Laplace Block.svg" src="Transfer_Functions_pliki/400px-Laplace_Block.png" width="400" height="94"></a></div>
</div>
<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=2" title="Edit section: Impulse Response">edit</a>]</span> <span class="mw-headline" id="Impulse_Response">Impulse Response</span></h2>
<div class="PrettyTextBox" style="float:right; clear:right; margin:1em; min-width:200px; width:250px; text-align:center;"><b>Note:</b>:<br>
Time domain variables are generally written with lower-case letters. 
Laplace-Domain, and other transform domain variables are generally 
written using upper-case letters.</div>
<p>For comparison, we will consider the time-domain equivalent to the 
above input/output relationship. In the time domain, we generally denote
 the input to a system as <i>x(t)</i>, and the output of the system as <i>y(t)</i>. The relationship between the input and the output is denoted as the <b>impulse response</b>, <i>h(t)</i>.</p>
<p>We define the impulse response as being the relationship between the 
system output to its input. We can use the following equation to define 
the impulse response:</p>
<dl>
<dd><img class="tex" alt="h(t) = \frac{y(t)}{x(t)}" src="Transfer_Functions_pliki/53e769b24657f28c1207caf090e6f86e.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=3" title="Edit section: Impulse Function">edit</a>]</span> <span class="mw-headline" id="Impulse_Function">Impulse Function</span></h3>
<p>It would be handy at this point to define precisely what an "impulse" is. The <b>Impulse Function</b>, denoted with δ<i>(t)</i> is a special function defined piece-wise as follows:</p>
<p><br style="clear: all;">
<span id="Impulse_Function"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Impulse Function]</p>
</div>
<dl>
<dd><img class="tex" alt="\delta(t) = \left\{
\begin{matrix} 
  0, &amp; t &lt; 0
\\
  \mbox{undefined}, &amp; t = 0
\\
  0, &amp; t &gt; 0
\end{matrix}\right.
" src="Transfer_Functions_pliki/32fd0759dce36780094d90ae1cbc69a9.png"></dd>
</dl>
<p>The impulse function is also known as the <b>delta function</b> 
because it's denoted with the Greek lower-case letter δ. The delta 
function is typically graphed as an arrow towards infinity, as shown 
below:</p>
<div class="center">
<div class="floatnone"><a href="http://en.wikibooks.org/wiki/File:Delta_Function.svg" class="image"><img alt="Delta Function.svg" src="Transfer_Functions_pliki/351px-Delta_Function.png" width="351" height="266"></a></div>
</div>
<p>It is drawn as an arrow because it is difficult to show a single 
point at infinity in any other graphing method. Notice how the arrow 
only exists at location 0, and does not exist for any other time <i>t</i>. The delta function works with regular time shifts just like any other function. For instance, we can graph the function δ<i>(t - N)</i> by shifting the function δ<i>(t)</i> to the right, as such:</p>
<div class="center">
<div class="floatnone"><a href="http://en.wikibooks.org/wiki/File:DeltaN_Function.svg" class="image"><img alt="DeltaN Function.svg" src="Transfer_Functions_pliki/351px-DeltaN_Function.png" width="351" height="266"></a></div>
</div>
<p>An examination of the impulse function will show that it is related to the unit-step function as follows:</p>
<dl>
<dd><img class="tex" alt="\delta(t) = \frac{du(t)}{dt}" src="Transfer_Functions_pliki/3369e7a1a0ff331cdaa75c304dd6dbab.png"></dd>
</dl>
<p>and</p>
<dl>
<dd><img class="tex" alt="u(t) = \int \delta(t) dt" src="Transfer_Functions_pliki/242eedaa52079661351365b019aa285e.png"></dd>
</dl>
<p>The impulse function is not defined at point <i>t = 0</i>, but the impulse response must always satisfy the following condition, or else it is not a true impulse function:</p>
<dl>
<dd><img class="tex" alt="\int_{-\infty}^\infty \delta(t)dt = 1" src="Transfer_Functions_pliki/2462eaa93ac7268f9a311f36db8a57ea.png"></dd>
</dl>
<p>The response of a system to an impulse input is called the <b>impulse response</b>.
 Now, to get the Laplace Transform of the impulse function, we take the 
derivative of the unit step function, which means we multiply the 
transform of the unit step function by s:</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}[u(t)] = U(s) = \frac{1}{s}" src="Transfer_Functions_pliki/671eebf71494450d1a3d1a02e29c5e11.png"></dd>
</dl>
<dl>
<dd><img class="tex" alt="\mathcal{L}[\delta(t)] = sU(s) = \frac{s}{s} = 1" src="Transfer_Functions_pliki/fdc1d771e8581c30ef5ab87ff5f50f74.png"></dd>
</dl>
<p>This result can be verified in the transform tables in <b><a href="http://en.wikibooks.org/wiki/Control_Systems/Transforms" title="Control Systems/Transforms">The Appendix</a></b>.</p>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=4" title="Edit section: Step Response">edit</a>]</span> <span class="mw-headline" id="Step_Response">Step Response</span></h3>
<div class="PrettyTextBox" style="clear: right; margin:0 0 1em 1em; font-size:90%; width: 250px; padding:4px; spacing:0px;text-align:left;float:right;">This operation can be performed using this <a href="http://en.wikibooks.org/wiki/MATLAB_Programming" title="MATLAB Programming">MATLAB</a> command:<br>
<center><b><a href="http://en.wikibooks.org/wiki/Control_Systems/MATLAB" title="Control Systems/MATLAB">step</a></b></center>
</div>
<p>Similarly to the impulse response, the <b>step response</b> of a 
system is the output of the system when a unit step function is used as 
the input. The step response is a common analysis tool used to determine
 certain metrics about a system. Typically, when a new system is 
designed, the step response of the system is the first characteristic of
 the system to be analyzed.</p>
<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=5" title="Edit section: Convolution">edit</a>]</span> <span class="mw-headline" id="Convolution">Convolution</span></h2>
<div class="PrettyTextBox" style="clear: right; margin:0 0 1em 1em; font-size:90%; width: 250px; padding:4px; spacing:0px;text-align:left;float:right;">This operation can be performed using this <a href="http://en.wikibooks.org/wiki/MATLAB_Programming" title="MATLAB Programming">MATLAB</a> command:<br>
<center><b><a href="http://en.wikibooks.org/wiki/Control_Systems/MATLAB" title="Control Systems/MATLAB">conv</a></b></center>
</div>
<p>However, the impulse response cannot be used to find the system 
output from the system input in the same manner as the transfer 
function. If we have the system input and the impulse response of the 
system, we can calculate the system output using the <b>convolution operation</b> as such:</p>
<dl>
<dd><img class="tex" alt="y(t) = h(t) * x(t)" src="Transfer_Functions_pliki/ca45ed01415f949551821bd19ba569d9.png"></dd>
</dl>
<div class="PrettyTextBox" style="float:right; clear:right; margin:1em; min-width:200px; width:250px; text-align:center;">Remember: an asterisk means <b>convolution</b>, not <b>multiplication</b>!</div>
<p>Where " * " (asterisk) denotes the convolution operation. Convolution
 is a complicated combination of multiplication, integration and 
time-shifting. We can define the convolution between two functions, <i>a(t)</i> and <i>b(t)</i> as the following:</p>
<p><br style="clear: all;">
<span id="Convolution"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Convolution]</p>
</div>
<dl>
<dd><img class="tex" alt="(a*b)(t) = (b*a)(t) = \int_{-\infty}^\infty a(\tau)b(t - \tau)d\tau" src="Transfer_Functions_pliki/05b21f13d5cf8625eab28cb3cef5d8fe.png"></dd>
</dl>
<p>(The variable τ (Greek tau) is a dummy variable for integration). 
This operation can be difficult to perform. Therefore, many people 
prefer to use the Laplace Transform (or another transform) to convert 
the convolution operation into a multiplication operation, through the <b>Convolution Theorem</b>.</p>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=6" title="Edit section: Time-Invariant System Response">edit</a>]</span> <span class="mw-headline" id="Time-Invariant_System_Response">Time-Invariant System Response</span></h3>
<p>If the system in question is time-invariant, then the general 
description of the system can be replaced by a convolution integral of 
the system's impulse response and the system input. We can call this the
 <b>convolution description</b> of a system, and define it below:</p>
<p><br style="clear: all;">
<span id="Convolution_Description"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Convolution Description]</p>
</div>
<dl>
<dd><img class="tex" alt="y(t) = x(t) * h(t) = \int_{-\infty}^\infty x(\tau)h(t - \tau)d\tau" src="Transfer_Functions_pliki/4d8e291d4e3124532d36392e00d4b434.png"></dd>
</dl>
<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=7" title="Edit section: Convolution Theorem">edit</a>]</span> <span class="mw-headline" id="Convolution_Theorem">Convolution Theorem</span></h2>
<p>This method of solving for the output of a system is quite tedious, 
and in fact it can waste a large amount of time if you want to solve a 
system for a variety of input signals. Luckily, the Laplace transform 
has a special property, called the <b>Convolution Theorem</b>, that makes the operation of convolution easier:</p>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<dl>
<dt>Convolution Theorem</dt>
<dd>Convolution in the time domain becomes multiplication in the complex
 Laplace domain. Multiplication in the time domain becomes convolution 
in the complex Laplace domain.</dd>
</dl>
</div>
</div>
<p>The Convolution Theorem can be expressed using the following equations:</p>
<p><br style="clear: all;">
<span id="Convolution_Theorem"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Convolution Theorem]</p>
</div>
<dl>
<dd><img class="tex" alt="\mathcal{L}[f(t) * g(t)] = F(s)G(s)" src="Transfer_Functions_pliki/c314eb8cb194e52ffe7f2bbe4da04a27.png"></dd>
<dd><img class="tex" alt="\mathcal{L}[f(t)g(t)] = F(s) * G(s)" src="Transfer_Functions_pliki/e50c044e4134563c428db78d0ed8c410.png"></dd>
</dl>
<p>This also serves as a good example of the property of <b>Duality</b>.</p>
<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=8" title="Edit section: Using the Transfer Function">edit</a>]</span> <span class="mw-headline" id="Using_the_Transfer_Function">Using the Transfer Function</span></h2>
<p>The Transfer Function fully describes a control system. The Order, 
Type and Frequency response can all be taken from this specific 
function. Nyquist and Bode plots can be drawn from the open loop 
Transfer Function. These plots show the stability of the system when the
 loop is closed. Using the denominator of the transfer function, called 
the characteristic equation the roots of the system can be derived.</p>
<p>For all these reasons and more, the Transfer function is an important
 aspect of classical control systems. Let's start out with the 
definition:</p>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<dl>
<dt>Transfer Function</dt>
<dd>The Transfer function of a system is the relationship of the 
system's output to its input, represented in the complex Laplace domain.</dd>
</dl>
</div>
</div>
<p>If the complex Laplace variable is <i>s</i>, then we generally denote the transfer function of a system as either <i>G(s)</i> or <i>H(s)</i>. If the system input is <i>X(s)</i>, and the system output is <i>Y(s)</i>, then the transfer function can be defined as such:</p>
<dl>
<dd><img class="tex" alt="H(s) = \frac{Y(s)}{X(s)}" src="Transfer_Functions_pliki/12a4dbad64576cc24327f6da5187ad06.png"></dd>
</dl>
<p>If we know the input to a given system, and we have the transfer 
function of the system, we can solve for the system output by 
multiplying:</p>
<p><br style="clear: all;">
<span id="Transfer_Function_Description"></span></p>
<div style="float: right; clear: all; margin-right: 30px; font-size: 90%; font-weight: bold;">
<p>[Transfer Function Description]</p>
</div>
<dl>
<dd><img class="tex" alt="Y(s) = H(s)X(s)" src="Transfer_Functions_pliki/b038557f6b66391c5799046c2c951720.png"></dd>
</dl>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=9" title="Edit section: Example: Impulse Response">edit</a>]</span> <span class="mw-headline" id="Example:_Impulse_Response">Example: Impulse Response</span></h3>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<p>From a Laplace transform table, we know that the Laplace transform of the impulse function, δ<i>(t)</i> is:</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}[\delta (t) ] = 1" src="Transfer_Functions_pliki/2c29b4334ca1329c543526c65cdb7cfd.png"></dd>
</dl>
<p>So, when we plug this result into our relationship between the input, output, and transfer function, we get:</p>
<dl>
<dd><img class="tex" alt="Y(s) = X(s)H(s)" src="Transfer_Functions_pliki/e9b598b8a0687e5a86b167455e9a0747.png"></dd>
<dd><img class="tex" alt="Y(s) = (1) H(s)" src="Transfer_Functions_pliki/b6544b05905cc9e4ff3e2a8ebf1ddd0d.png"></dd>
<dd><img class="tex" alt="Y(s) = H(s)" src="Transfer_Functions_pliki/3248517a586fdc5b2198b2d0d9b30b1e.png"></dd>
</dl>
<p>In other words, the "impulse response" is the output of the system when we input an impulse function.</p>
</div>
</div>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=10" title="Edit section: Example: Step Response">edit</a>]</span> <span class="mw-headline" id="Example:_Step_Response">Example: Step Response</span></h3>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<p>From the Laplace Transform table, we can also see that the transform of the unit step function, <i>u(t)</i> is given by:</p>
<dl>
<dd><img class="tex" alt="\mathcal{L}[u(t)] = \frac{1}{s}" src="Transfer_Functions_pliki/cef3da08f2a1bd8183b38d6f67b3dfa0.png"></dd>
</dl>
<p>Plugging that result into our relation for the transfer function gives us:</p>
<dl>
<dd><img class="tex" alt="Y(s) = X(s)H(s)" src="Transfer_Functions_pliki/e9b598b8a0687e5a86b167455e9a0747.png"></dd>
<dd><img class="tex" alt="Y(s) = \frac{1}{s}H(s)" src="Transfer_Functions_pliki/2bdbe159dca71c287d8c539451a40f80.png"></dd>
<dd><img class="tex" alt="Y(s) = \frac{H(s)}{s}" src="Transfer_Functions_pliki/20e70398639e1de7b45d958750e400e5.png"></dd>
</dl>
<p>And we can see that the step response is simply the impulse response divided by <i>s</i>.</p>
</div>
</div>
<h3><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=11" title="Edit section: Example: MATLAB Step Response">edit</a>]</span> <span class="mw-headline" id="Example:_MATLAB_Step_Response">Example: MATLAB Step Response</span></h3>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<p>Use MATLAB to find the step response of the following transfer function:</p>
<dl>
<dd><img class="tex" alt="F(s)=\frac{79s^2+916s+1000}{s(s+10)^3}" src="Transfer_Functions_pliki/545cdd997834af921c30b702e2e2b86f.png"></dd>
</dl>
<p>We can separate out our numerator and denominator polynomials as such:</p>
<pre>num = [79 916 1000];
den = [1 30 300 1000 0];
sys = tf(num, den);
</pre>
<p>Now, we can get our step response from the <b>step</b> function, and plot it for time from 1 to 10 seconds:</p>
<pre>T = 1:0.001:10;
step(sys, T);
</pre></div>
</div>
<h2><span class="editsection">[<a href="http://en.wikibooks.org/w/index.php?title=Control_Systems/Transfer_Functions&amp;action=edit&amp;section=12" title="Edit section: Frequency Response">edit</a>]</span> <span class="mw-headline" id="Frequency_Response">Frequency Response</span></h2>
<p>The <b>Frequency Response</b> is similar to the Transfer function, 
except that it is the relationship between the system output and input 
in the complex Fourier Domain, not the Laplace domain. We can obtain the
 frequency response from the transfer function, by using the following 
change of variables:</p>
<dl>
<dd><img class="tex" alt="s = j\omega" src="Transfer_Functions_pliki/a85b8a883b07d53a9bf9d8c408de5c5c.png"></dd>
</dl>
<div style="clear:both; width:90%; margin:5px auto; padding:0px;">
<div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;">
<dl>
<dt>Frequency Response</dt>
<dd>The frequency response of a system is the relationship of the system's output to its input, represented in the Fourier Domain.</dd>
</dl>
</div>
</div>
<div class="center">
<div class="floatnone"><a href="http://en.wikibooks.org/wiki/File:Fourier_Block.svg" class="image"><img alt="Fourier Block.svg" src="Transfer_Functions_pliki/400px-Fourier_Block.png" width="400" height="92"></a></div>
</div>
<p>Because the frequency response and the transfer function are so 
closely related, typically only one is ever calculated, and the other is
 gained by simple variable substitution. However, despite the close 
relationship between the two representations, they are both useful 
individually, and are each used for different purposes.</p>
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</td>
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</td>
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<p><a href="http://en.wikibooks.org/wiki/Control_Systems/Sampled_Data_Systems" title="Control Systems/Sampled Data Systems"><font color="white">Sampled Data Systems →</font></a></p>
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